Analytical calculation of the parameters of light bullets propagating in the tunnel ionization regime

Laser Physics ◽  
2021 ◽  
Vol 32 (2) ◽  
pp. 025401
Author(s):  
Aleksandr N Bugay ◽  
Vyacheslav A Khalyapin
Keyword(s):  
Differential Equations ◽  
Frequency Shift ◽  
Ordinary Differential Equations ◽  
Stationary Solution ◽  
Analytical Calculation ◽  
Temporal Delay ◽  
Stable Regime ◽  
Transparent Dielectrics ◽  
Light Bullets ◽  
Analytic Estimation

Abstract Analytic estimation of the parameters of light bullets formed in the anomalous group dispersion region of transparent dielectrics under conditions of tunneling photoionization was performed. For this purpose, the system of the ordinary differential equations for the laser pulse’s parameters such as amplitude, temporal duration, chirp parameter, temporal delay, frequency shift, radius and curvature were obtained. The stationary solution of this system and conditions of the quasi-stable regime of propagation were found.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

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